I have spent some time being confused by the nature of global methods in number theory. It seems that there are in some sense (for my purposes) three levels at which algebraic number theorists operate: local (at one prime), everywhere local (at all primes simultaneously, including the infinite ones) and global (actually playing with the number field). When we talk about things like local class field theory we mean the first one, and when we talk about global class field theory I guess we mean the last one (but the extent to which the ideles seem to get involved suggests to my naive mind a strong whiff of the second one also).

When we talk about local to global principles we most definitely mean the passage from the second to the third. I guess my question can therefore by crystallised in terms of elementary number theory as follows: what global techniques do we have for proving the non-existence of solutions to diophantine equations? In other words, given a failure of the local-global principle, what are the techniques one can use to demonstrate it independently of local information?

If I am given a diophantine equation and asked to show it has no solutions, I can think of very few methods that are not in some sense `local', certainly if we count working at the infinite primes also as local (which surely we should?).

One possibility I have been considering is that something to do with heights/descent is perhaps a global method. However, the height of a point is still measurable by concatenating local data, and descent is normally via some trick involving congruences, but perhaps the `well-orderedness' of the process is a truly global trick.

Also, returning to a pessimistic analysis, in an important classical result that I have seen called a `measure of the failure of the local-global principle': the finiteness of the class number, it seems to me that the statement doesn't involve the infinite primes in any way, so to prove it we milk the infinite primes for all the have got and get the Minkowski bound by directly studying local behaviour above infinity. Is my opinion in this regard incorrect? If so, which of the arguments are truly global?

So to conclude, are there such things as "global methods", and if there are, what are they? Apologies for posing what is probably a naive, overly-simplistic and absurdly general question, but I am hoping several people may have thought about this and have interesting things to say.

You say finiteness of the class number doesn't use the infinite primes in its proof. Not so! You use the embedding of the number field into a Euclidean space (via all real and complex places) to view the ring of integers and its nonzero ideals as lattices, to which you apply Minkowski's theorem. As for an example of a "global" method to show a Diophantine equation has no integral solutions, there are methods using factorization, descent, and/or units. For example, to prove $y^2 = x^3 + 6$ has no integral solutions, you can use unique factorization in ${\mathbf Z}[\sqrt{6}]$ with its units.
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KConradMay 30 '12 at 21:31

This technique does use congruences at some point, but the key idea is to work with prime factorization in a particular ring, and you must know what all the units are to make progress.
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KConradMay 30 '12 at 21:32

I see now that in your comment on the class number you do point out that its proof involves arguments at infinity.
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KConradMay 30 '12 at 21:33

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The equation $y^2 = x^3 - 51$ has no integral solution, but it has real and $p$-adic solutions for all $p$ (with the $p$-adic solutions being $p$-adic integers). To prove there are no integral solutions, even though there are local solutions at all places, you use the fact that the class number is 2. Methods that involve class numbers and (global) units are global methods, not local methods. (By the way, there are rational solutions.)
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KConradMay 30 '12 at 21:40

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You claim that the adeles $\mathbb{A}_K$ of a global field $K$ are "everywhere local simultaneously" rather than global. Perhaps this is true, but once you start thinking about how $K$ sits inside $\mathbb{A}_K$, then you are definitely in a truly global setting.
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David LoefflerMay 31 '12 at 6:52

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The following example is perhaps expressed a bit loosely but it is morally correct, and it is fundamental.

Suppose that for each prime $p$ we have chosen a decomposition group $G_p=\mathrm{Gal}(\bar{\mathbf{Q}}_p|\mathbf{Q}_p)\ $ of $G=\mathrm{Gal}(\bar{\mathbf{Q}}|\mathbf{Q})$. If we are given a global character $\chi:G\to\mathbf{C}^\times$, then we get by restriction a family of local characters $\chi_p$ of $G_p$ almost all of which are unramified in the sense of being trivial on the inertia subgroup $I_p\subset G_p$. Conversely, when does a family $(\chi_p)_p$ of local characters, almost all of which are unramified, come from a global character ?

The reciprocity isomorphism of local class field theory at the various primes allows us to attach a character of finite order $\xi_p:\mathbf{Q}_p^\times\to\mathbf{C}^\times\ $ to each $\chi_p$. Since the $\chi_p$ are almost all unramified, the $\xi_p$ are almost all trivial on $\mathbf{Z}_p^\times$, and hence give rise to a character $\xi:\mathbf{A}^\times\to\mathbf{C}^\times$ of the idèles of $\mathbf{Q}$. Now the condition for the $\chi_p$ to come from a global $\chi:G\to\mathbf{C}^\times$ is that this $\xi$ should be trivial on the subgroup $\mathbf{Q}^\times\subset\mathbf{A}^\times$. In other words, $\xi$ should come from a character $\mathbf{A}^\times/\mathbf{Q}^\times\to\mathbf{C}^\times$. This is a truly global condition and cannot be expressed by any collection of local conditions.

I suggest a course of treatment with Mordell's book Diophantine Equations. It is perhaps an extreme case, but it represents number theory "before Bourbaki got his hands on it", as a leading British mathematician once was heard to say. Bare-hands methods are typically as global as you like.

Bourbaki never "got his hands" on Number Theory. Where does he treat quadratic reciprocity, or the density of primes in arithmetic progressions, or the analytic continuation and functional equation for the zeta function, or the maximal abelian extension of $\mathbf{Q}$, or the maximal unramified extension of $\mathbf{Q}$, or the cusp form of level 1 and weight 12 ?
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Chandan Singh DalawatMay 31 '12 at 4:43

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That's being pedantic - I know what was meant and it was the Langlands program. And/or the Antwerp Modular Forms Conference 1972. The intellectual consequences of Borel and Serre's involvement in the Seminar on Complex Multiplication.
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Charles MatthewsMay 31 '12 at 6:58

I know the book. And it starts with a discussion on the question of whether integral solutions of homogeneous systems are morally the same as rational points on the underlying projective variety, an interesting and puzzling point (until you realise that the issue must be about parametric families of solutions). I gave a historical answer to a methodological question, which may seem perverse. But "history written by the victors" plays a major role in talking about methods.
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Charles MatthewsMay 31 '12 at 7:43

This is quite a brief answer as I am not quite sure your question is appropriately phrased for this site, but one can use the "Brauer-Manin obstruction" to show the non-existence of rational points, even if the variety is everywhere locally soluble. Try searching google - in particular papers by Colliot-Thélène, Sansuc, Harari, Swinnerton-Dyer, Skorobogatov and many others.

I'm not quite sure if this classifies as "truely global" according to your definition, as the method works by cutting out a certain subset of the adèles which contains the set of rational points, and the adèles are built out of all the completions of the number field.

However in my opinion I would certainly say that it is a global method and the general set-up of how it works relies on results from global class field theory.

This is probably far-fetched, but the Weil-Shimura-Taniyama conjecture was a global conjecture (now solved), whose proof implied the non-existence of solutions for a famous Diophantine equation $x^n+y^n=z^n$ for sufficiently large $n$.

Showing that someting is automorphic requires some general tools from the Langlands program, such as the Arthur trace formula or converse theorems. The latter require analytic continuation, functional equations and boundedness in vertical stripes, which are purely global features.

The Arthur trace formula or the more specialized Eichler-Selberg trace formula are also a global construction, which requires an understanding of all local places, and glue this information together in terms of the conjugacy classes, of say $GL_2(\mathbb{Q})$. It allows you to match automorphic coeffecients (Hecke eigenvalues) with arithmetic coeffecients (I guess: these are the eigenvalues of an action on the cohomology theory) of the Hasse Weil Zeta function, as soon as you find a good geometric realization of the cohomology theory for the later.